Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C23×C4 |
| direct product | C127×#12573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #99 |
| commutator | C23 | C2×C42×C127 |
| Frattini, cosocle | C22×D4 | C23×C127 |
| socle | C22×C127 | C2×#99 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 147 points:
(5,6)(7,11)(8,15,16,10)(9,12)(13,19,20,17)(14,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,2,3,4)(6,10)(11,17)(12,15)(18,19);(5,7,9,14)(6,11,12,18)(8,13,16,20)(10,17,15,19)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5561);